Illuminations: Paper Pool

Paper Pool

Unit Overview

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Which Pocket?

In this lesson, students continue their investigation by discovering a rule to predict the pocket in which the ball will land. As an extension, students can also consider the number of squares that a ball crosses while traversing its path.

Learning Objectives

Students will:

Identify patterns that occur in the Paper Pool game
Predict the pocket in which the ball will land, given the dimensions of a table

Materials

Computer with Internet connection
Paper Pool Record Sheet or Paper Pool Recording Spreadsheet
Crossing Squares Activity Sheet
Advanced Paper Pool Activity Sheet (optional)

Instructional Plan

Inform students that they will use the data they've already collected to investigate another outcome: the pocket in which the ball lands.

Allow students to use the data that they collected in previous lessons. If the data is organized in the Paper Pool Record Sheet, students may be able to make some observations. Alternatively, if the data is recorded in the Paper Pool Excel Spreadsheet, students can sort the data. Sorting the data first by "Pocket" and then by "Number of Hits" may give results like the following:

Length (horizontal)	Height (vertical)	Number of Hits	Pocket
2	1	3	B
4	2	3	B
2	3	5	B
2	5	7	B
4	5	9	B
4	4	2	C
1	1	2	C
3	1	4	C
5	1	6	C
1	7	8	C
5	3	8	C
1	9	10	C
3	7	10	C
3	2	5	D
1	6	7	D

Seeing the data organized in this way allows students to look for patterns within subcategories, such as considering only the tables for which the ball lands in pocket C.

Ask students to note observations about the subcategories. Three big observations that students may suggest are the following:

If the ball lands in pocket B, the length is even.
If the ball lands in pocket D, the height is even.
If the ball lands in pocket C, then either:
- the length and height are equal, or
- the length and height are both odd.

Further investigation will allow students to conjecture the following rule:

If the dimensions, reduced to lowest terms, are... The ball lands in pocket...

Even × Odd B

Odd × Odd C

Odd × Even D

Note that if the dimensions are Even × Even, then each dimension can be divided by (at least) a factor of 2, and the dimenions will reduce to one of cases above.

The lessons in this unit were arranged so that students could determine a pattern for the number of hits independently of determining the pocket in which the ball will land. However, you may wish to have students search for both patterns simultaneously, in which case you would and to combine the How Many Hits?, Graphical Representations, and Which Pocket? lessons.

Assessment Options

At the end of this unit, students will write a final report, which is described in the Wrapping It Up lesson plan. For this report, students should save their notes and records from this lesson. You may want to look at each student's work during the class to assess their current level of understanding and identify areas where help is needed, but sutdents should be responsible for holding onto their notes and keeping them organized.

Once students identify rules for the number of hits and the pocket in which the ball will land, they can answer the questions on the Advanced Paper Pool activity sheet.

Advanced Paper Pool Activity Sheet

Extensions

Students can consider the number of squares that the ball crosses while traveling across a pool table. Students can record their results on the Crossing Squares activity sheet, or they can add another row to the Paper Pool Excel Spreadsheet file.

Crossing Squares Activity Sheet

The number of squares that the ball crosses is given by the following expression:

Number of Squares = (l × h) ÷ GCF(l,h) = LCM(l,h)

That is, the number of squares crossed is equal to the leaast common multiple of the length (l) and height (h).

The paths that the ball takes can be investigated. Although all paths are horizontally, vertically, or rotationally symmetric, students may be interested to learn that the pattern depends on the dimensions of the table. [If the ball lands in pocket B, the path is horizontally symmetric; if the ball lands in pocket D, the path is vertically symmetric; and, if the ball lands in pocket C, the path is rotationally symmetric.]

Teacher Reflection

What strategies would be helpful for students to more easily identify patterns regarding the pocket? What questions could you ask to suggest those strategies, without giving too big a hint?
Were students able to organize information on their own, or did they need the structure of a prepared record sheet?
What observations did students make that you did not expect? How did you deal with student suggestions—did you allow time for exploration, or did you promise to return to their idea in a future lesson?

NCTM Standards and Expectations

Algebra 6-8

Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Data Analysis & Probability 6-8

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

References

Lappan, Glenda, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips. "Comparing and Scaling: Ratio, Proportion, and Percent," in Connected Mathematics Project. Upper Saddle River, NJ: Pearson Prentice Hall, 2004.